Representing solids with nets and views

A closed cube has six square faces. What does one of those squares represent?

Count surfaces, joins and corners separately

A face is a flat surface of a polyhedron. Two faces meet along an edge, and edges meet at vertices

This closed cuboid has six rectangular faces, twelve edges and eight vertices

Count hidden edges and corners as well as those at the front. A dashed line in this drawing shows an edge behind a face

A cube is a special cuboid with all six faces equal squares

A drawing lies on a flat page, even when it represents a three-dimensional object. Do not measure apparent sloping lengths or angles unless the drawing is stated to be to scale

Unfold the surface without adding or losing a face

A net is a flat arrangement of faces that folds to make the surface of a solid

Imagine cutting selected edges of a closed box while keeping its faces connected, then opening them flat

To construct it, draw each face with matching edge lengths, cut only the outside boundary and fold the shared edges

The folds must bring the unjoined boundary edges together without overlapping faces or leaving a hole

In the net diagrams here, dashed shared edges are folds. Their meaning differs from the hidden-edge dashes in the assembled cuboid drawing

Six equal connected squares are necessary for these cube nets, but are not sufficient. Their arrangement must place each square on a different cube face

Follow six labels to six different cube faces

The six squares have equal side lengths and join along full edges

Cut out only the outside boundary and fold along the dashed shared edges. Do not detach or overlap faces

Can these stated folds make a closed cube without overlap?

Keep one square as a starting face and fold the connected squares at right angles

With A held at the front, following the shared folds places A at the front; B at the top; C at the back; D at the right; E at the bottom; F at the left

Track each labelled face rather than judging only the flat outline

The opposite pairs after folding are A and C; B and E; D and F

All six squares occupy different face positions, covering the cube without overlap

Track a differently arranged strip of faces

Six squares can still fold onto occupied positions

The six squares have equal side lengths and join along full edges

Cut out only the outside boundary and fold along the dashed shared edges. Do not detach or overlap faces

Can these stated folds make a closed cube without overlap?

Follow the chain of squares and fold each shared edge through a right angle

With A held at the front, following the shared folds places A at the front; B at the top; C at the back; D at the bottom; E at the front; F at the top

Faces A and E would occupy the same face position

Faces B and F would occupy the same face position

The overlapping faces cannot both cover that position

Other face positions remain uncovered, so this is not a net of a closed cube

Check the face positions rather than the square count

Another valid outline can cover the same cube

The six squares have equal side lengths and join along full edges

Cut out only the outside boundary and fold along the dashed shared edges. Do not detach or overlap faces

Can these stated folds make a closed cube without overlap?

Keep one square as a starting face and fold the connected squares at right angles

With A held at the front, following the shared folds places A at the front; B at the top; C at the right; D at the back; E at the left; F at the bottom

Track each labelled face rather than judging only the flat outline

The opposite pairs after folding are A and D; B and F; C and E

All six squares occupy different face positions, covering the cube without overlap

Test a compact arrangement with shared-edge constraints

Adjacency in the flat net is not the full cube relationship

Adjacent cube faces share an edge after folding. Opposite faces point away from each other and do not share an edge

Choose one labelled square as a starting face. Follow each shared edge and turn the next square through a right angle

A straight chain of four squares wraps around four faces; its first and third faces are opposite, as are its second and fourth

Track attached flaps as well. They must cover the two remaining face positions

Every cube face has one opposite face. List three opposite pairs to account for all six faces

Two squares that do not touch in the flat net are not necessarily opposite after folding

List the three opposite pairs in a valid net

The six squares have equal side lengths and join along full edges

This is a valid cube net. Fold along the dashed shared edges and keep the face labels attached to their squares

Which labelled face is opposite A after folding?

Keep one square as a starting face and fold the connected squares at right angles

With A held at the front, following the shared folds places A at the front; B at the top; C at the back; D at the right; E at the bottom; F at the left

Track each labelled face rather than judging only the flat outline

The opposite pairs after folding are A and C; B and E; D and F

All six squares occupy different face positions, covering the cube without overlap

The face opposite A is C

Find the opposite face after the bent strip folds

Distinguish an opposite face from every adjacent face

Match two end faces and the strip around them

A triangular prism has two congruent triangular end faces and three rectangular side faces

The three rectangles form a strip. Their widths match the three triangle sides a, b and c, while their common length L is the prism length

Here the triangle sides are 3 cm, 4 cm and 5 cm, and the prism length is 6 cm

To draw this net accurately, construct the three rectangles with widths 3 cm, 4 cm and 5 cm and common length 6 cm, then attach matching triangles at the shown opposite strip edges

Folding the strip joins its two outer edges; the two triangles close the ends

A rectangular-prism net instead has six rectangular faces in three matching opposite pairs. A cube is the equal-square special case

Distinguish one base and meeting triangular faces

This square pyramid has one square base and four triangular side faces

Each triangle is attached to one square edge. Folding raises the triangles until their tips meet at one apex

Unlike a prism, this pyramid has no second matching base at the other end

To draw this particular net, make a square with side 4 cm and attach an isosceles triangle to each side with perpendicular face height 5 cm

The face height is measured within a triangular face, not vertically through the assembled pyramid

The net shows every face without hiding one behind another, which helps when planning a cut-and-fold model

A proposed closed triangular-prism net contains three rectangles and only one triangle. All joins match, but no extra face will be attached. Can it close the whole prism?

Read a pictorial drawing without treating it as a photograph

A parallel-projection drawing keeps edges that run in the same spatial direction parallel on the page

An isometric drawing is a particular parallel projection with equal scale along three main directions, often drawn using vertical and two thirty-degree grid directions

To sketch a box on an isometric grid, mark its width, depth and height along those directions, then complete matching parallel edges

A perspective drawing instead makes receding parallel edges tend towards vanishing points. It gives a view-dependent appearance

Either pictorial drawing can show several faces at once, but faces or edges behind the object may be hidden

The displayed cuboid is a schematic parallel projection, not a measured isometric grid or a perspective drawing. Its written dimensions, not its apparent screen lengths, state the measurements

Look straight from one fixed direction

An orthographic view looks straight at an object from one direction without perspective convergence

A front view shows horizontal width and vertical height, but collapses depth

A right-side view shows depth horizontally and height vertically. A top view shows the footprint from above

Keep front, back, left and right fixed to the object. Moving to the back does not rename the object sides

Label each view so the reader knows its direction and which way is up

To draw a view on squared paper, project each occupied position straight towards the viewing plane and shade its unit square. Do not draw sloping depth lines in a front elevation

Keep the height fixed and change depth first. Then keep depth fixed and change height. Compare which front, right and top view dimensions change. The pictorial box remains four units wide

Which observation is valid for every allowed setting?

Use a height plan to specify every cube

A height plan gives the number of unit cubes stacked at each base position

In these tasks, every stack starts on the same flat base and has no gaps or overhangs, so the plan specifies the whole object

Zero means the base position is empty. A two means two cubes stacked vertically at that position

The printed plan has back above front and left before right. These are object directions, not directions that turn with the viewer

For a front or side elevation, compare stack heights along each line of sight. Use the tallest stack, not their sum

Enter the view as a grid code: 1 for an occupied unit square and 0 for an empty unit square. Complete every entry, including zeros

The grid has a labelled upper row for height two and a lower row for height one. It encodes the same occupied squares you would shade in a drawing

Compare front and back heights in each column

An object is built from equal unit cubes standing on a flat two-by-two base grid

Each plan number gives the height of the stack at that position; zero means no cubes there. All stacks start on the base, with no gaps or overhangs

Front, back, left and right refer to the plan labels and stay fixed when the viewing direction changes

Complete the front view as a two-by-two grid code

Enter 1 for each unit square occupied in that view and 0 for an empty square. Complete all four entries

Look horizontally from the front; output columns are left then right

Look straight from the front; do not count cubes along the depth direction as extra height

The output columns run left then right

Left column: compare stack heights 1 and 0; the tallest is 1

Right column: compare stack heights 2 and 1; the tallest is 2

Put 1 in each unit square up to that height, and 0 above it

Use the upper row for height two and the lower row for height one

A back stack can appear above a shorter front stack

Look across left and right stacks from the right

An object is built from equal unit cubes standing on a flat two-by-two base grid

Each plan number gives the height of the stack at that position; zero means no cubes there. All stacks start on the base, with no gaps or overhangs

Front, back, left and right refer to the plan labels and stay fixed when the viewing direction changes

Complete the right view as a two-by-two grid code

Enter 1 for each unit square occupied in that view and 0 for an empty square. Complete all four entries

Look horizontally from the right; output columns are front then back

Look straight from the right; do not count cubes along the depth direction as extra height

The output columns run front then back

Front column: compare stack heights 2 and 0; the tallest is 2

Back column: compare stack heights 1 and 1; the tallest is 1

Put 1 in each unit square up to that height, and 0 above it

Use the upper row for height two and the lower row for height one

Keep front before back in the right-side output

Reverse the horizontal order when looking from the opposite side

A back view uses the same tallest heights as the front view, but reverses the horizontal order

From the back, the object right side appears in the left column of the page. The output is therefore labelled Right then Left

A left-side view similarly reverses the front-to-back order used in the right-side view

Its output columns are labelled Back then Front

For example, front-view column heights 1 then 2 become back-view column heights 2 then 1

The two-row codes are then front: upper row 0, 1 and lower row 1, 1; back: upper row 1, 0 and lower row 1, 1

Read the output headings rather than copying the original plan left to right

Construct the back view with its labelled column order

Construct the left view without reversing height

A footprint records positions, not stack heights

Looking straight down shows whether each base position is occupied

A stack of one cube and a stack of two cubes both cover one unit square in the top view

Replace every positive height by 1 and keep each zero as 0

For a bottom view of these gap-free stacks, the same base positions are occupied, but the page orientation must be stated

Here back stays at the top of the page. Looking up from below then places the object right side in the left page column

The bottom-view columns are therefore Right then Left, while the top-view columns are Left then Right

Convert positive heights to occupied footprint squares

An object is built from equal unit cubes standing on a flat two-by-two base grid

Each plan number gives the height of the stack at that position; zero means no cubes there. All stacks start on the base, with no gaps or overhangs

Front, back, left and right refer to the plan labels and stay fixed when the viewing direction changes

Complete the top view as a two-by-two grid code

Enter 1 for each unit square occupied in that view and 0 for an empty square. Complete all four entries

Keep the back row at the top of the page; the columns are left then right

Look straight down from above or straight up from below, as requested

A base position is occupied in this view if at least one cube stands there

Keep back at the top of the page; the requested top view has columns left then right

Back row: 0, 1

Front row: 1, 1

The footprint alone does not show the stack heights

Use the stated bottom-view orientation

Two closed cuboids each have front width 4 units and height 3 units. One has depth 1 unit and the other depth 2 units. Can an unlabelled front view alone distinguish them?

You need to cut one flat piece of card and fold it into a closed box. Which representation most directly shows the connected faces and fold edges to draw on the card?

Choose the representation and state what it leaves out

A net shows all faces and their folding connections, but not the assembled appearance directly

Track all six cube face positions: face count alone does not rule out overlaps or conflicting folds

A pictorial drawing suggests three-dimensional appearance, but can hide faces and should not be measured as though it were a scale orthographic view

An orthographic view shows two dimensions without perspective convergence and collapses the direction of sight

Construct elevation grid codes using the tallest stack on each sight line; construct top and bottom footprints using occupied base positions

Keep the requested view labels and orientation. Opposite viewing directions can reverse the page order

Several views or a fully specified height plan can reveal information that one view omits